Question

Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".

Answer 1

I have typeset Deligne's letter, and placed the result here:

http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf

I have made some minor edits so that the text reads more naturally to a native speaker of English. Also I made a few annotations where I truly believe there is an error in the original. Any other errors are mine.

The letter struck me with how much it accomplishes in such a short space. Actually, I would say that the meat of the argument is confined to the final three pages, with the rest there only to establish notation. This letter ought to be required reading for anyone studying automorphic forms in arithmetic!

Answer 2

I have a scan that I made of the copy in Kevin Buzzard's office some years ago, but I probably shouldn't post it without permission. It's also not the most legible copy - I suspect it is many generations of photocopies old. If someone can get Deligne's permission I'd be happy to make the PDF available - but I bet there are other people here with better copies, in any case.

EDIT: I now have permission, so here is my scan.

[removed]

It's possible that a clearer scan will be forthcoming; I will update this post if so.

EDIT: here is an improved scan, from Deligne's own copy.

http://dl.dropbox.com/u/66812/deligne-pish-improvedscan.pdf

Answer 3

I think it was published obscurely and in a Russian transcription (the original is in English) .

In endnote 25 of the review of Harris-Taylor on his website here, Milne writes:

[Deligne proved it] in an eleven-page handwritten letter to Piatetskii-Shapiro dated March 25, 1973, with a one-page typed covering letter dated April, 1973: “In it I claim (except at 2) to prove for the supercuspidal representations what in your notes [Antwerp Conference LNM 349] you prove for the principal (unramified) series. The idea is that

1. room is left for it in your notes only thanks to the supersingular elliptic curves;
2. supersingular elliptic curves correspond to ideal classes in the quaternion algebra ramified at p and 1;
3. this, by a global argument using Jacquet Langlands, forces the outcome.”

Apparently (see MR 50 7095), the letter was published: Matematika — Period. Sb. Perevodov Inostrannykh Statei, 18 (1974), 110–122. It would be useful if someone would put it on the web, since it was the starting point for Carayol and H&T. In fact, Deligne’s proof of (b) was completed by J-L. Brylinski (appendix to Carayol 1986).

---

Below is the complete typed covering letter.

Bures-sur-Yvette, April 2, 1973.

Dear Piatetski-Shapiro [the name is handwritten in Russian]

I am ashamed the enclosed letter is so badly written, and written more for my sake than for yours. In it, I claim (except at 2) to prove for the supersingular cuspidal representations what in your notes you prove for the principal (unramified serie [sic]. The idea is that

a) room is left for it in your notes only thanks to the supersingular elliptic curves;

b) supersingular elliptic curves correspond to ideal classes in the quaternion algebra ramified at $p$ and $\infty$;

c) this, by a global argument using Jacquet Langlands \S 14, forces the outcome

Corollary 1: Let $E$ be an elliptic curve $/\mathbb{Q}$, which is a direct factor (up to isogeny) of a jacobian of a modular curve, corresponding to a new form $\omega_E$. Then, except perhaps at $2$, the conductors of $E$ and $\omega_E$ are the same (At 2, I still can prove $2|f_E\iff 2|f_{\omega_E}$)

Corollary 2: (Casselman-Morita): Let $H$ be a quaternion algebra over $\mathbb{Q}$ split at $\infty$, $p$ be a prime at which $H$ ramifies, $\mathcal{O}$ be an order of $H$ maximal at $p$ and $M=X/\mathcal{O}^*$ [$X=$ Poincar\'e upper half plane]. Then, the irreducible components of the reduction of $M$ mod $p$ are rational. [$M$ is defined over a cyclotomic field unramified at $p$].

Mumford has nice ideas about the compactification of $K\backslash G/\Gamma$ ($K\backslash G$ hermitian symmetric). For the moduli of abelian varieties, I hope it will eventually give a nice compactification over $\mathbb{Z}[\frac{1}{\textrm{obvious primes}}]$.

I am looking forward to meeting you again.

Yours most sincerely,

P. DELIGNE

Answer 4

Find it here. (Edit: Link removed). I hope you can read Russian. Enjoy!

Edit: Find here another (better?) scan, also in Russian.